Semi-supervised Active Regression

Abstract

Labelled data often comes at a high cost as it may require recruiting human labelers or running costly experiments. At the same time, in many practical scenarios, one already has access to a partially labelled, potentially biased dataset that can help with the learning task at hand. Motivated by such settings, we formally initiate a study of $semi-supervised$ $active$ $learning$ through the frame of linear regression. In this setting, the learner has access to a dataset $X \in \mathbb{R}^{(n_1+n_2) \times d}$ which is composed of $n_1$ unlabelled examples that an algorithm can actively query, and $n_2$ examples labelled a-priori. Concretely, denoting the true labels by $Y \in \mathbb{R}^{n_1 + n_2}$, the learner's objective is to find $\widehat{\beta} \in \mathbb{R}^d$ such that, \begin{equation} \| X \widehat{\beta} - Y \|_2^2 \le (1 + \epsilon) \min_{\beta \in \mathbb{R}^d} \| X \beta - Y \|_2^2 \end{equation} while making as few additional label queries as possible. In order to bound the label queries, we introduce an instance dependent parameter called the reduced rank, denoted by $R_X$, and propose an efficient algorithm with query complexity $O(R_X/\epsilon)$. This result directly implies improved upper bounds for two important special cases: (i) active ridge regression, and (ii) active kernel ridge regression, where the reduced-rank equates to the statistical dimension, $sd_\lambda$ and effective dimension, $d_\lambda$ of the problem respectively, where $\lambda \ge 0$ denotes the regularization parameter. For active ridge regression we also prove a matching lower bound of $O(sd_\lambda / \epsilon)$ on the query complexity of any algorithm. This subsumes prior work that only considered the unregularized case, i.e., $\lambda = 0$.